论文标题
CM等级扭曲图的2个图像
2-adic images of CM isogeny-torsion graphs
论文作者
论文摘要
令$ \ mathcal {e} $为$ \ mathbb {q} $ - 在$ \ mathbb {q} $上定义的椭圆曲线的同基因类别。与$ \ Mathcal {e} $相关的同学图是一个图表，每个元素的顶点$ \ nathcal {e} $，每个$ \ m athbb {q} $的优势 - 优质学位的等值范围 - 将一个椭圆形曲线映射为$ \ nathcal {e} $ apt $ \ mathcal {e} $ \ nath $ nthe $ \ nthe $ \ \等级式记录为边缘标签。与$ \ MATHCAL {E} $关联的等级扭曲图是与$ \ Mathcal {E} $相关的等级图，此外，我们将每个顶点标记为$ \ Mathbb {Q} $的抽象组结构，该结构的抽象组结构是相应的Elliptic Elliptic curve的。本文的主要结果是对等值折叠图的每个顶点的2个galois图像的分类，其相关的$ \ mathbb {q} $ - 同类等级类别由超过$ \ mathbb {q} $的椭圆曲线组成，并具有复杂的乘法。
Let $\mathcal{E}$ be a $\mathbb{Q}$-isogeny class of elliptic curves defined over $\mathbb{Q}$. The isogeny graph associated to $\mathcal{E}$ is a graph which has a vertex for each element of $\mathcal{E}$ and an edge for each $\mathbb{Q}$-isogeny of prime degree that maps one elliptic curve in $\mathcal{E}$ to another elliptic curve in $\mathcal{E}$, with the degree of the isogeny recorded as a label of the edge. The isogeny-torsion graph associated to $\mathcal{E}$ is the isogeny graph associated to $\mathcal{E}$ where, in addition, we label each vertex with the abstract group structure of the torsion subgroup over $\mathbb{Q}$ of the corresponding elliptic curve. The main result of the article is a classification of the 2-adic Galois image at each vertex of the isogeny-torsion graphs whose associated $\mathbb{Q}$-isogeny class consists of elliptic curves over $\mathbb{Q}$ with complex multiplication.
